Use the cosine rule in a non-right-angled triangle

Use the angle between the two known sides

Angle A is opposite side a and lies between sides b and c. SAS determines the remaining side. With all three positive sides satisfying the triangle inequalities, rearrange to find the angle

1. 1SSS rearrangement Use the angle opposite side a \(\displaystyle \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\)

Two sides have lengths 5 cm and 8 cm with included angle 60°. Find the square of the opposite side

Known sides

\(\displaystyle 5\,\mathrm{cm},\quad8\,\mathrm{cm}\)

Included angle

\(\displaystyle 60^\circ\)

Find the side opposite the 60° angle, in cm

For sides 5,5,8, find the cosine of the angle opposite 8.

For a triangle with sides 5, 5 and 8, the cosine of the angle opposite 8 is −7/25. What does this tell you about the angle?

Carry the base side into the solid

For a triangular horizontal base with adjacent sides 5 and 8 and included angle 60°, the opposite base side is 7. A vertical height attached at its endpoint is perpendicular to that whole base plane, so it can be combined with that side in a right section. Keep the exact 7, rather than an unnecessarily rounded intermediate value

Justify the cosine rule, ungraded

Resolve one side into components parallel and perpendicular to another, then use Pythagoras. Explain why a negative projection handles an obtuse included angle

Model: signed projection

Place the included-angle vertex at the origin and one side of length c along a horizontal axis. Resolve the other side into horizontal and vertical components, then use the squared distance between the endpoints. If A is obtuse, its horizontal component is negative, so the same identity applies